Chapter 7 - Section 7.4 - Integration of Rational Functions by Partial Fractions. - 7.4 Exercises - Page 515: 26

$\frac{1}{2} + \ln 6$

$$\eqalign{ & \int_1^2 {\frac{{{x^3} + 4{x^2} + x - 1}}{{{x^3} + {x^2}}}} dx \cr & {\text{The degree of the numerator is greater than the degree of the }} \cr & {\text{denominator}},{\text{ we first perform the long division}} \cr & \frac{{{x^3} + 4{x^2} + x - 1}}{{{x^3} + {x^2}}} = 1 + \frac{{3{x^2} + x - 1}}{{{x^3} + {x^2}}} \cr & {\text{The partial fraction decomposition of }}\frac{{3{x^2} + x - 1}}{{{x^3} + {x^2}}}{\text{ has the}} \cr & {\text{form}} \cr & \frac{{3{x^2} + x - 1}}{{{x^3} + {x^2}}} = \frac{{3{x^2} + x - 1}}{{{x^2}\left( {x + 1} \right)}} = \frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{x + 1}} \cr & {\text{Multiply both sides by the least common denominator}} \cr & 3{x^2} + x - 1 = Ax\left( {x + 1} \right) + B\left( {x + 1} \right) + C{x^2} \cr & 3{x^2} + x - 1 = A{x^2} + Ax + Bx + B + C{x^2} \cr & {\text{Collecting like terms}} \cr & 3{x^2} + x - 1 = \left( {A{x^2} + C{x^2}} \right) + \left( {Ax + Bx} \right) + B \cr & {\text{We obtain the following system of equations}} \cr & A + C = 3 \cr & A + B = 1 \cr & B = - 1 \cr & {\text{Solving}} \cr & A = 2,{\text{ }}B = - 1,{\text{ }}C = 1 \cr & {\text{Then}} \cr & \frac{{3{x^2} + x - 1}}{{{x^2}\left( {x + 1} \right)}} = \frac{A}{x} + \frac{B}{{{x^2}}} + \frac{C}{{x + 1}} \cr & \frac{{3{x^2} + x - 1}}{{{x^2}\left( {x + 1} \right)}} = \frac{2}{x} - \frac{1}{{{x^2}}} + \frac{1}{{x + 1}} \cr & \frac{{{x^3} + 4{x^2} + x - 1}}{{{x^3} + {x^2}}} = 1 + \frac{2}{x} - \frac{1}{{{x^2}}} + \frac{1}{{x + 1}} \cr & \int_1^2 {\frac{{{x^3} + 4{x^2} + x - 1}}{{{x^3} + {x^2}}}} dx = \int_1^2 {\left( {1 + \frac{2}{x} - \frac{1}{{{x^2}}} + \frac{1}{{x + 1}}} \right)} dx \cr & {\text{Integrating}} \cr & {\text{ = }}\left[ {x + 2\ln \left| x \right| + \frac{1}{x} + \ln \left| {x + 1} \right|} \right]_1^2 \cr & {\text{Evaluating}} \cr & {\text{ = }}\left[ {2 + 2\ln \left| 2 \right| + \frac{1}{2} + \ln \left| {2 + 1} \right|} \right] - \left[ {1 + 2\ln \left| 1 \right| + \frac{1}{1} + \ln \left| {1 + 1} \right|} \right] \cr & {\text{ = }}\left( {2 + 2\ln 2 + \frac{1}{2} + \ln 3} \right) - \left( {2 + \ln 2} \right) \cr & {\text{ = }}\frac{5}{2} + 2\ln 2 + \ln 3 - 2 - \ln 2 \cr & {\text{ = }}\frac{1}{2} + \ln 2 + \ln 3 \cr & {\text{ = }}\frac{1}{2} + \ln 6 \cr} $$